Question: Solve for $X$. $\left[\begin{array}{rr}5 & 5 & -7 \\ 2 & 3 & -6 \end{array}\right]+X=\left[\begin{array}{rr}1 & 1 & 5 \\ 4 & -2 & 9\end{array}\right] $ $X=$
Solution: The Strategy First, we can represent the matrices of the equation with letters, which will make the equation easier to handle. Then we can solve the equation for $X$ and obtain an expression with the letters we defined. Finally, we can substitute back the actual matrices into the resulting expression and simplify it. Solving the equation for $X$ We are given the following equation. $\left[\begin{array}{rr}5 & 5 & -7 \\ 2 & 3 & -6 \end{array}\right]+X=\left[\begin{array}{rr}1 & 1 & 5 \\ 4 & -2 & 9\end{array}\right] $ Let's represent the above matrices as follows. $A=\left[\begin{array}{rr}5 & 5 & -7 \\ 2 & 3 & -6 \end{array}\right] ~~~~~~~~~ B = \left[\begin{array}{rr}1 & 1 & 5 \\ 4 & -2 & 9\end{array}\right]$ Then we can rewrite the equation as follows. $A+X=B$ Now it's simple to solve the equation for $X$. $\begin{aligned}A+X&=B\\\\ X&=B-A\end{aligned}$ Finding $X$ We found that $X=B-A$. Now we can substitute the actual matrices back into the expression and simplify. $\begin{aligned}X&=B-A \\\\&=\left[\begin{array}{rr}1 & 1 & 5 \\ 4 & -2 & 9\end{array}\right]-\left[\begin{array}{rr}5 & 5 & -7 \\ 2 & 3 & -6 \end{array}\right] \\\\\\&=\left[\begin{array}{rr}(1-5) & (1-5) & (5+7) \\ (4-2) & (-2-3) & (9+6)\end{array}\right] \\\\\\&=\left[\begin{array}{rr}-4 & -4 & 12 \\ 2 & -5 & 15\end{array}\right]\end{aligned}$ Summary $X=\left[\begin{array}{rr}-4 & -4 & 12 \\ 2 & -5 & 15\end{array}\right]$